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PROFESSOR: Last time we left off
with a question having to

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00:00:24,840 --> 00:00:26,185
do with playing with blocks.

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00:00:26,185 --> 00:00:29,960
And this is supposed to give
us a visceral feel for

12
00:00:29,960 --> 00:00:32,820
something anyway, having
to do with series.

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00:00:32,820 --> 00:00:35,650
And the question was whether
I could stack these blocks,

14
00:00:35,650 --> 00:00:39,460
build up a stack so that I'm
going to try here up.

15
00:00:39,460 --> 00:00:42,920
I'm already off balance
here, see.

16
00:00:42,920 --> 00:00:48,410
The question is can I build
this so that the--

17
00:00:48,410 --> 00:00:53,380
let's draw a picture of
it so that the first

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00:00:53,380 --> 00:00:54,870
block is like this.

19
00:00:54,870 --> 00:00:56,800
The next block is like this.

20
00:00:56,800 --> 00:00:58,150
And maybe the next block
is like this.

21
00:00:58,150 --> 00:01:00,340
And notice there is no
visible means of

22
00:01:00,340 --> 00:01:02,410
support for this block.

23
00:01:02,410 --> 00:01:08,310
It's completely to the left
of the first block.

24
00:01:08,310 --> 00:01:11,055
And the question is, will
this fall down?

25
00:01:11,055 --> 00:01:13,730

26
00:01:13,730 --> 00:01:21,290
Or at least, or more precisely,
eventually we'll

27
00:01:21,290 --> 00:01:24,650
ask you know, how
far can we go?

28
00:01:24,650 --> 00:01:30,050
Now before you answer this
question, the claim is it that

29
00:01:30,050 --> 00:01:33,480
this is a kind of a natural,
physical question, which

30
00:01:33,480 --> 00:01:37,330
involves some important
answer.

31
00:01:37,330 --> 00:01:39,410
No matter whether the
answer is, you can

32
00:01:39,410 --> 00:01:40,290
do it or you can't.

33
00:01:40,290 --> 00:01:43,080
So this is a good kind of math
question where no matter what

34
00:01:43,080 --> 00:01:45,455
the answer is, when you figure
out the answer, you're going

35
00:01:45,455 --> 00:01:47,350
to get something interesting
out of it.

36
00:01:47,350 --> 00:01:48,570
Because they're two
possibilities.

37
00:01:48,570 --> 00:01:53,490
Either there is a limit to how
far to the left we can go-- in

38
00:01:53,490 --> 00:01:56,400
which case that's a very
interesting number--

39
00:01:56,400 --> 00:01:58,810
or else there is no limit.

40
00:01:58,810 --> 00:02:00,370
You can go arbitrarily far.

41
00:02:00,370 --> 00:02:03,240
And that's also interesting
and curious.

42
00:02:03,240 --> 00:02:05,540
And that's the difference
between convergence and

43
00:02:05,540 --> 00:02:10,100
divergence, the thing that we
were talking about up to now

44
00:02:10,100 --> 00:02:11,380
concerning series.

45
00:02:11,380 --> 00:02:17,440
So my first question is, do you
think that I can get it so

46
00:02:17,440 --> 00:02:21,470
that this thing doesn't fall
down with, well you see I have

47
00:02:21,470 --> 00:02:24,020
about eight blocks here or so.

48
00:02:24,020 --> 00:02:25,570
So you can vote now.

49
00:02:25,570 --> 00:02:28,550
How many in favor that I can
succeed in doing this sort of

50
00:02:28,550 --> 00:02:30,450
thing with maybe more
than three blocks.

51
00:02:30,450 --> 00:02:32,040
How many in favor?

52
00:02:32,040 --> 00:02:33,860
All right somebody
is voting twice.

53
00:02:33,860 --> 00:02:34,560
That's good.

54
00:02:34,560 --> 00:02:35,980
I like that.

55
00:02:35,980 --> 00:02:37,230
How about opposed?

56
00:02:37,230 --> 00:02:39,930

57
00:02:39,930 --> 00:02:43,450
So that was really
close to a tie.

58
00:02:43,450 --> 00:02:44,780
All right.

59
00:02:44,780 --> 00:02:49,220
But I think the there was
slightly more opposed.

60
00:02:49,220 --> 00:02:49,570
I don't know.

61
00:02:49,570 --> 00:02:52,000
You guys who are in the
back maybe could tell.

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00:02:52,000 --> 00:02:53,810
Anyway it was pretty close.

63
00:02:53,810 --> 00:02:54,250
All right.

64
00:02:54,250 --> 00:02:55,220
So now I'm going--

65
00:02:55,220 --> 00:02:57,280
because this is a real
life thing--

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00:02:57,280 --> 00:02:58,200
I'm going to try to do it.

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All right?

68
00:02:59,476 --> 00:03:00,550
All right.

69
00:03:00,550 --> 00:03:04,240
So now I'm going to tell
you what the trick is.

70
00:03:04,240 --> 00:03:08,890
The trick is to do
it backwards.

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00:03:08,890 --> 00:03:12,120
When most people are playing
with blocks, they decide to

72
00:03:12,120 --> 00:03:14,420
build it from the bottom up.

73
00:03:14,420 --> 00:03:15,670
Right?

74
00:03:15,670 --> 00:03:19,280
But we're going to build
it from the top down,

75
00:03:19,280 --> 00:03:20,760
from the top down.

76
00:03:20,760 --> 00:03:22,820
And that's going to make it
possible for us to do the

77
00:03:22,820 --> 00:03:25,080
optimal thing at each stage.

78
00:03:25,080 --> 00:03:28,900
So when I build it from the top
down, the best I can do is

79
00:03:28,900 --> 00:03:30,520
well, it'll fall off.

80
00:03:30,520 --> 00:03:34,040
I need to have it you know,
halfway across.

81
00:03:34,040 --> 00:03:35,470
That's the best I can do.

82
00:03:35,470 --> 00:03:38,000
So the top one I'm going
to build like that.

83
00:03:38,000 --> 00:03:41,730
I'm going to take it as far
to the left as I can.

84
00:03:41,730 --> 00:03:44,640
And then I'm going to put the
next one down as far to the

85
00:03:44,640 --> 00:03:46,160
left as I can.

86
00:03:46,160 --> 00:03:50,450
And then the next one as far
to the left as I can.

87
00:03:50,450 --> 00:03:51,740
That was a little too far.

88
00:03:51,740 --> 00:03:54,900
And then I'm going to do
the next one as far to

89
00:03:54,900 --> 00:03:56,382
the left as I can.

90
00:03:56,382 --> 00:03:58,470
And then I'm going to do
the next one-- well

91
00:03:58,470 --> 00:04:01,720
let's line it up first--

92
00:04:01,720 --> 00:04:04,450
as far to the left as I can.

93
00:04:04,450 --> 00:04:05,150
OK?

94
00:04:05,150 --> 00:04:09,036
And then the next one as far
to the left as I can.

95
00:04:09,036 --> 00:04:09,920
All right.

96
00:04:09,920 --> 00:04:15,700
Now those of you who are in this
line can see, all right,

97
00:04:15,700 --> 00:04:16,650
I succeeded.

98
00:04:16,650 --> 00:04:18,740
All right, that's
over the edge.

99
00:04:18,740 --> 00:04:20,050
All right?

100
00:04:20,050 --> 00:04:21,604
So it can be done.

101
00:04:21,604 --> 00:04:22,854
All right.

102
00:04:22,854 --> 00:04:25,450

103
00:04:25,450 --> 00:04:26,590
All right.

104
00:04:26,590 --> 00:04:32,990
So now we know that we can get
farther than you know, we can

105
00:04:32,990 --> 00:04:34,500
make it overflow.

106
00:04:34,500 --> 00:04:37,390
So the question now is,
how far can I get?

107
00:04:37,390 --> 00:04:38,240
OK.

108
00:04:38,240 --> 00:04:40,630
Do you think I can
get to here?

109
00:04:40,630 --> 00:04:43,260
Can I get to the
end over here?

110
00:04:43,260 --> 00:04:50,170
So how many people think I can
get this far over to here?

111
00:04:50,170 --> 00:04:51,730
How many people think
I can get this far?

112
00:04:51,730 --> 00:04:53,390
Well you know, remember.

113
00:04:53,390 --> 00:04:56,340
I'm going to have to use more
than just this one more block

114
00:04:56,340 --> 00:04:56,560
that I've got.

115
00:04:56,560 --> 00:04:58,280
I don't, right?

116
00:04:58,280 --> 00:05:01,270
Obviously I'm thinking, actually
I do have some more

117
00:05:01,270 --> 00:05:02,330
blocks at home.

118
00:05:02,330 --> 00:05:03,370
But, OK.

119
00:05:03,370 --> 00:05:04,085
We're not going to.

120
00:05:04,085 --> 00:05:06,510
But anyway, do you think
I can get over to here?

121
00:05:06,510 --> 00:05:09,710
How many people say yes?

122
00:05:09,710 --> 00:05:13,060
And how many people say no?

123
00:05:13,060 --> 00:05:15,370
More people said no then yes.

124
00:05:15,370 --> 00:05:15,680
All right.

125
00:05:15,680 --> 00:05:18,680
So maybe the stopping place is
some mysterious number in

126
00:05:18,680 --> 00:05:19,995
between here.

127
00:05:19,995 --> 00:05:20,690
All right?

128
00:05:20,690 --> 00:05:21,720
Well OK.

129
00:05:21,720 --> 00:05:24,840
So now we're going to
do the arithmetic.

130
00:05:24,840 --> 00:05:26,010
And we're going to
figure out what

131
00:05:26,010 --> 00:05:28,870
happens with this problem.

132
00:05:28,870 --> 00:05:29,500
OK?

133
00:05:29,500 --> 00:05:32,175
So let's do it.

134
00:05:32,175 --> 00:05:36,780
All right, so now again the
idea is, the idea is we're

135
00:05:36,780 --> 00:05:44,210
going to start with the
top, the top block.

136
00:05:44,210 --> 00:05:46,710

137
00:05:46,710 --> 00:05:48,450
We'll call that block
number one.

138
00:05:48,450 --> 00:05:53,070

139
00:05:53,070 --> 00:05:56,620
And then the farthest if you
like to the right, that you

140
00:05:56,620 --> 00:05:59,070
can put a block underneath
it, is exactly halfway.

141
00:05:59,070 --> 00:06:01,800

142
00:06:01,800 --> 00:06:04,610
All right well, that's the
best job I can do.

143
00:06:04,610 --> 00:06:09,440
Now in order to make my units
work out easily, I'm going to

144
00:06:09,440 --> 00:06:13,160
decide to call the length
of the block 2.

145
00:06:13,160 --> 00:06:14,090
All right?

146
00:06:14,090 --> 00:06:17,800
And that means if I start at
location 0, then the first

147
00:06:17,800 --> 00:06:22,550
place where I am is supposed
to be halfway.

148
00:06:22,550 --> 00:06:24,070
And that will be 1.

149
00:06:24,070 --> 00:06:26,650

150
00:06:26,650 --> 00:06:30,070
OK so the first step
in the process is

151
00:06:30,070 --> 00:06:31,660
1 more to the right.

152
00:06:31,660 --> 00:06:34,040
Or if you like, if I were
building up-- which is what

153
00:06:34,040 --> 00:06:35,620
you would actually have
to do in real life--

154
00:06:35,620 --> 00:06:39,280
it would be 1 to the left.

155
00:06:39,280 --> 00:06:41,610
OK now the next one.

156
00:06:41,610 --> 00:06:44,960
Now here is the way that you
start figuring out the

157
00:06:44,960 --> 00:06:45,460
arithmetic.

158
00:06:45,460 --> 00:06:48,190
The next one is based on
a physical principle.

159
00:06:48,190 --> 00:06:53,350
Which is that the farthest I
can stick this next block

160
00:06:53,350 --> 00:06:57,800
underneath is what's called the
center of mass of these

161
00:06:57,800 --> 00:07:00,210
two, which is exactly
halfway here.

162
00:07:00,210 --> 00:07:02,970
That is there's 1/4 of this
guy, and a 1/4 of that guy

163
00:07:02,970 --> 00:07:04,660
balancing each other.

164
00:07:04,660 --> 00:07:04,900
Right?

165
00:07:04,900 --> 00:07:06,160
So that's as far as I can go.

166
00:07:06,160 --> 00:07:08,625
If I go farther than that,
it'll fall over.

167
00:07:08,625 --> 00:07:10,600
So that's the absolute
farthest I can do.

168
00:07:10,600 --> 00:07:14,630
So the next block is going
to be over here.

169
00:07:14,630 --> 00:07:18,210
And 1/4 of 2 is 1/2.

170
00:07:18,210 --> 00:07:21,552
So this is 3/2 here.

171
00:07:21,552 --> 00:07:24,430
All right so we went to 1.

172
00:07:24,430 --> 00:07:27,960
We went to it 3/2 here.

173
00:07:27,960 --> 00:07:31,300
And then I'm going to keep on
going with this eventually.

174
00:07:31,300 --> 00:07:34,790
All right so we're going to
figure out what happens with

175
00:07:34,790 --> 00:07:36,770
this stack.

176
00:07:36,770 --> 00:07:37,774
Question?

177
00:07:37,774 --> 00:07:40,302
AUDIENCE: How do you
know that this is

178
00:07:40,302 --> 00:07:42,670
the best way to optimize?

179
00:07:42,670 --> 00:07:44,590
PROFESSOR: The question is how
do I know that this is the

180
00:07:44,590 --> 00:07:46,870
best way to optimize?

181
00:07:46,870 --> 00:07:49,210
I can't answer that question.

182
00:07:49,210 --> 00:07:51,630
But I can tell you that it's the
best way if I start with a

183
00:07:51,630 --> 00:07:53,860
top like this, and the
next one like this.

184
00:07:53,860 --> 00:07:56,100
Right, because I'm doing
the farthest

185
00:07:56,100 --> 00:07:57,470
possible at each stage.

186
00:07:57,470 --> 00:08:00,130
That actually has a name in
computer science that's called

187
00:08:00,130 --> 00:08:01,610
the greedy algorithm.

188
00:08:01,610 --> 00:08:04,520
I'm trying to do the best
possible at each stage.

189
00:08:04,520 --> 00:08:07,310
The greedy algorithm starting
from the bottom, is an

190
00:08:07,310 --> 00:08:09,420
extremely bad strategy.

191
00:08:09,420 --> 00:08:13,130
Because when you do that, you
stack it this way, and it

192
00:08:13,130 --> 00:08:14,120
almost falls over.

193
00:08:14,120 --> 00:08:16,050
And then the next time you
can't do anything.

194
00:08:16,050 --> 00:08:19,030
So the greedy algorithm is
terrible from the bottom.

195
00:08:19,030 --> 00:08:21,850
This is the greedy algorithm
starting from the top, and it

196
00:08:21,850 --> 00:08:24,720
turns out to do much better
then the greedy algorithm

197
00:08:24,720 --> 00:08:25,580
starting from the bottom.

198
00:08:25,580 --> 00:08:27,390
But of course I'm not addressing
whether there might

199
00:08:27,390 --> 00:08:30,420
not be some other incredibly
clever strategy where I wiggle

200
00:08:30,420 --> 00:08:34,260
around and make it go up.

201
00:08:34,260 --> 00:08:35,676
I'm not addressing
that question.

202
00:08:35,676 --> 00:08:36,130
All right?

203
00:08:36,130 --> 00:08:37,650
It turns out this is the
best you can do.

204
00:08:37,650 --> 00:08:40,790
But that's not clear.

205
00:08:40,790 --> 00:08:44,650
All right so now, here
we have this thing.

206
00:08:44,650 --> 00:08:47,460
And now I have to figure out
what the arithmetic pattern

207
00:08:47,460 --> 00:08:48,850
is, so that I can figure
out what I was

208
00:08:48,850 --> 00:08:50,340
doing with those shapes.

209
00:08:50,340 --> 00:08:53,850

210
00:08:53,850 --> 00:08:58,132
So let's figure out a thought
experiment here.

211
00:08:58,132 --> 00:08:59,570
All right?

212
00:08:59,570 --> 00:09:03,040
Now the thought experiment I
want to imagine for you is,

213
00:09:03,040 --> 00:09:08,550
you've got a stack of a bunch
of blocks, and this is the

214
00:09:08,550 --> 00:09:10,070
first N blocks.

215
00:09:10,070 --> 00:09:14,150

216
00:09:14,150 --> 00:09:15,310
All right?

217
00:09:15,310 --> 00:09:19,010
And now we're going to put
one underneath it.

218
00:09:19,010 --> 00:09:25,230
And what we're going to figure
out is the center of mass of

219
00:09:25,230 --> 00:09:30,960
those N blocks, which I'm going
to call C sub N. OK.

220
00:09:30,960 --> 00:09:32,920
And that's the place where
I'm going to put

221
00:09:32,920 --> 00:09:34,320
this very next block.

222
00:09:34,320 --> 00:09:36,480
I'll put it in a different
color here.

223
00:09:36,480 --> 00:09:39,560
Here's the new, the
next block over.

224
00:09:39,560 --> 00:09:42,360
And the next block over is
the N plus 1st block.

225
00:09:42,360 --> 00:09:49,900

226
00:09:49,900 --> 00:09:54,380
And now I want you to think
about what's going on here.

227
00:09:54,380 --> 00:09:58,150
If the center of mass of the
first N blocks is this number,

228
00:09:58,150 --> 00:10:01,700
this new one, it's
of length 2.

229
00:10:01,700 --> 00:10:05,720
And its center of mass is 1
further to the right than the

230
00:10:05,720 --> 00:10:07,850
center of mass that
we had before.

231
00:10:07,850 --> 00:10:10,580
So in other words, I've added
to this configuration of N

232
00:10:10,580 --> 00:10:13,220
blocks, 1 more block,
which is shifted.

233
00:10:13,220 --> 00:10:16,560
Whose center mass is not lined
up with the center of mass of

234
00:10:16,560 --> 00:10:20,200
this, but actually over
farther to the right.

235
00:10:20,200 --> 00:10:25,280
All right so the new center of
mass of this new block--

236
00:10:25,280 --> 00:10:27,060
and this is the extra
piece of information

237
00:10:27,060 --> 00:10:28,440
that I want to observe--

238
00:10:28,440 --> 00:10:36,740
is that this thing has a center
of mass at C N plus 1.

239
00:10:36,740 --> 00:10:39,870
It's 1 unit over because
this total length is 2.

240
00:10:39,870 --> 00:10:43,390
So right in the middle
there is 1 over,

241
00:10:43,390 --> 00:10:45,561
according to my units.

242
00:10:45,561 --> 00:10:48,580
All right now this is going to
make it possible for me to

243
00:10:48,580 --> 00:10:51,970
figure out what the new
center of mass is.

244
00:10:51,970 --> 00:11:04,860
So C N plus 1 is the center of
mass of N plus 1 blocks.

245
00:11:04,860 --> 00:11:08,910
Now this is really only in the
horizontal variable, right?

246
00:11:08,910 --> 00:11:10,860
I'm not keeping track of
the center of mass.

247
00:11:10,860 --> 00:11:13,105
Actually this thing is hard to
build because the center of

248
00:11:13,105 --> 00:11:14,240
mass is also rising.

249
00:11:14,240 --> 00:11:15,830
It's getting higher
and higher.

250
00:11:15,830 --> 00:11:18,500
But I'm only keeping track
of its left and right

251
00:11:18,500 --> 00:11:19,510
characteristic.

252
00:11:19,510 --> 00:11:21,590
So this is the x-coordinate
of it.

253
00:11:21,590 --> 00:11:27,760

254
00:11:27,760 --> 00:11:31,590
All right so now here's
the idea.

255
00:11:31,590 --> 00:11:35,920
I'm combining the white ones,
the N blocks, with the pink

256
00:11:35,920 --> 00:11:37,630
one, which is the one
on the bottom.

257
00:11:37,630 --> 00:11:41,020
And there are N of
the white ones.

258
00:11:41,020 --> 00:11:42,520
And there's 1 of the pink one.

259
00:11:42,520 --> 00:11:44,520
And so in order to get the
center of mass of the whole, I

260
00:11:44,520 --> 00:11:47,700
have to take the weighted
average of the two.

261
00:11:47,700 --> 00:11:55,270
That's N times C N plus 1, times
the center of mass of

262
00:11:55,270 --> 00:12:00,270
the pink one, which
is C N plus 1.

263
00:12:00,270 --> 00:12:02,095
And then I have to divide if
it's the weighted average of

264
00:12:02,095 --> 00:12:06,560
the total of N plus 1
blocks, by N plus 1.

265
00:12:06,560 --> 00:12:09,440
This is going to give me the
new center of mass of my

266
00:12:09,440 --> 00:12:11,855
configuration at the
N plus 1st stage.

267
00:12:11,855 --> 00:12:14,800

268
00:12:14,800 --> 00:12:16,660
And now I can just do the
arithmetic and figure

269
00:12:16,660 --> 00:12:19,340
out what this is.

270
00:12:19,340 --> 00:12:21,810
And the two C Ns combine.

271
00:12:21,810 --> 00:12:30,250
I get N plus 1, times C N plus
1, divided by N plus 1.

272
00:12:30,250 --> 00:12:32,620
And if I combine these
two things and do the

273
00:12:32,620 --> 00:12:36,140
cancellation, that gives me this
recurrence formula, C N

274
00:12:36,140 --> 00:12:41,040
plus 1 is equal to C N plus,
there's a little extra.

275
00:12:41,040 --> 00:12:42,010
These two cancel.

276
00:12:42,010 --> 00:12:44,240
That gives me the C N.
But then I also have

277
00:12:44,240 --> 00:12:46,740
1 over N plus 1.

278
00:12:46,740 --> 00:12:54,230

279
00:12:54,230 --> 00:12:57,270
Well that's how much gain I can
get in the center of mass

280
00:12:57,270 --> 00:12:58,620
by adding one more block.

281
00:12:58,620 --> 00:13:02,270
That's how much I can shift
things over, depending on how

282
00:13:02,270 --> 00:13:05,130
we're thinking of things to
the left or the right,

283
00:13:05,130 --> 00:13:07,137
depending on which direction
we're building them.

284
00:13:07,137 --> 00:13:10,960

285
00:13:10,960 --> 00:13:13,840
All right, so now I'm going
to work out the formulas.

286
00:13:13,840 --> 00:13:17,680
First of all C 1, that was the
center of the first block.

287
00:13:17,680 --> 00:13:19,920
I put its left-end at
0, the center of the

288
00:13:19,920 --> 00:13:22,250
first block is at 1.

289
00:13:22,250 --> 00:13:25,910
That means that C 1 is 1.

290
00:13:25,910 --> 00:13:27,310
OK?

291
00:13:27,310 --> 00:13:30,550
C 2, according to
this formula--

292
00:13:30,550 --> 00:13:32,470
and actually I've worked it
out, we'll check it in a--

293
00:13:32,470 --> 00:13:35,530
C 2 is C 1 plus 1 over 2.

294
00:13:35,530 --> 00:13:37,380
All right, so that's the
case, N equals 1.

295
00:13:37,380 --> 00:13:40,130
So this is 1 plus 1/2.

296
00:13:40,130 --> 00:13:43,420
That's what we already did.

297
00:13:43,420 --> 00:13:45,790
That's the 3/2 number.

298
00:13:45,790 --> 00:13:51,320
Now the next one is
C sub 2 plus 1/3.

299
00:13:51,320 --> 00:13:53,530
That's the formula again.

300
00:13:53,530 --> 00:13:59,590
And so that comes out to
be 1 plus 1/2 plus 1/3.

301
00:13:59,590 --> 00:14:02,740
And now you can see what
the pattern is.

302
00:14:02,740 --> 00:14:08,650
C N, if you just keep on going
here, C N is going to be 1

303
00:14:08,650 --> 00:14:21,410
plus 1/2 plus 1/3 plus
1/4 plus 1/N.

304
00:14:21,410 --> 00:14:25,600
So now I would like
you to vote again.

305
00:14:25,600 --> 00:14:27,790
Do you think I can-- now that
we have the formula--

306
00:14:27,790 --> 00:14:30,830
do you think I can
get over to here?

307
00:14:30,830 --> 00:14:33,180
How many people think I
can get over to here?

308
00:14:33,180 --> 00:14:36,160

309
00:14:36,160 --> 00:14:40,410
How many people think I can't
get over to here?

310
00:14:40,410 --> 00:14:42,700
There's still a lot
of people who do.

311
00:14:42,700 --> 00:14:46,420
So it's still almost 50/50.

312
00:14:46,420 --> 00:14:47,470
That's amazing.

313
00:14:47,470 --> 00:14:49,800
Well so we'll address that
in a few minutes.

314
00:14:49,800 --> 00:14:52,690
So now let me tell you
what's going on.

315
00:14:52,690 --> 00:14:55,420
This C N of course, is the same
as what we called last

316
00:14:55,420 --> 00:15:00,070
time S N. And remember that
we actually estimated the

317
00:15:00,070 --> 00:15:01,230
size of this guy.

318
00:15:01,230 --> 00:15:04,620
This is related to what's called
the harmonic series.

319
00:15:04,620 --> 00:15:10,300
And what we showed was that log
N is less than S N, which

320
00:15:10,300 --> 00:15:14,690
is less than S N plus 1.

321
00:15:14,690 --> 00:15:15,940
All right?

322
00:15:15,940 --> 00:15:17,910

323
00:15:17,910 --> 00:15:23,150
Now I'm going to call your
attention to the red part,

324
00:15:23,150 --> 00:15:26,710
which is the divergence part
of this estimate, which is

325
00:15:26,710 --> 00:15:30,290
this one for the time
being, all right.

326
00:15:30,290 --> 00:15:32,780
Just saying that this
thing is growing.

327
00:15:32,780 --> 00:15:41,270
And what this is saying is that
as N goes to infinity,

328
00:15:41,270 --> 00:15:52,230
log N goes to infinity, So that
means that S N goes to

329
00:15:52,230 --> 00:15:57,980
infinity, because of this
inequality here.

330
00:15:57,980 --> 00:16:02,130
It's bigger than log N. And so
if N is big enough, we can get

331
00:16:02,130 --> 00:16:04,002
as far as we like.

332
00:16:04,002 --> 00:16:06,030
All right?

333
00:16:06,030 --> 00:16:08,530
So I can get to here.

334
00:16:08,530 --> 00:16:11,030
And at least half of you, at
least the ones who voted, that

335
00:16:11,030 --> 00:16:12,330
was I don't know.

336
00:16:12,330 --> 00:16:13,890
We have a quorum here,
but I'm not sure.

337
00:16:13,890 --> 00:16:16,470
We certainly didn't have a
majority on either side.

338
00:16:16,470 --> 00:16:19,420
Anyway this thing does
go to infinity.

339
00:16:19,420 --> 00:16:21,900
So in principle, if I had enough
blocks, I could get it

340
00:16:21,900 --> 00:16:25,298
over to here.

341
00:16:25,298 --> 00:16:26,920
All right, and that's
the meaning of

342
00:16:26,920 --> 00:16:28,300
divergence in this case.

343
00:16:28,300 --> 00:16:32,430

344
00:16:32,430 --> 00:16:36,850
On the other hand, I want to
discuss with you, and the

345
00:16:36,850 --> 00:16:39,840
reason why I use this example,
is I want to discuss with you

346
00:16:39,840 --> 00:16:45,700
also what's going on with this
other inequality here, and

347
00:16:45,700 --> 00:16:49,360
what its significance is.

348
00:16:49,360 --> 00:16:53,020
Which is that it's going to take
us a lot of numbers N, a

349
00:16:53,020 --> 00:16:57,050
lot of blocks, to get up
to a certain level.

350
00:16:57,050 --> 00:16:59,350
In other words, I can't
do it with just eight

351
00:16:59,350 --> 00:17:00,800
blocks or nine blocks.

352
00:17:00,800 --> 00:17:02,300
In order to get over here,
I'd have to use

353
00:17:02,300 --> 00:17:06,000
quite a few of them.

354
00:17:06,000 --> 00:17:08,620
So let's just see
how many it is.

355
00:17:08,620 --> 00:17:11,120

356
00:17:11,120 --> 00:17:14,230
So I worked this
out carefully.

357
00:17:14,230 --> 00:17:15,645
And let's see what I got.

358
00:17:15,645 --> 00:17:18,150

359
00:17:18,150 --> 00:17:31,760
So to get across the lab
tables, all right.

360
00:17:31,760 --> 00:17:35,780
This distance here, I already
did this secretly.

361
00:17:35,780 --> 00:17:38,985
But I don't actually even have
enough of these to show you.

362
00:17:38,985 --> 00:17:43,850
But, well 1, 2, 3,
4, 5, 6, and 1/2.

363
00:17:43,850 --> 00:17:44,750
I guess that's enough.

364
00:17:44,750 --> 00:17:46,240
So it's 6 and 1/2.

365
00:17:46,240 --> 00:17:50,000
So it's two lab tables is
13 of these blocks.

366
00:17:50,000 --> 00:17:51,190
All right.

367
00:17:51,190 --> 00:18:00,310
So there are 13 blocks, which
is equal to 26 units.

368
00:18:00,310 --> 00:18:04,020
OK, that's how far to
get across I need.

369
00:18:04,020 --> 00:18:06,090
And the first one
is already 2.

370
00:18:06,090 --> 00:18:09,470
So it's really 26 minus
2, which is 24.

371
00:18:09,470 --> 00:18:11,450
Which that's what I need.

372
00:18:11,450 --> 00:18:13,110
OK.

373
00:18:13,110 --> 00:18:23,240
So I need log N to be equal to
24, roughly speaking, in order

374
00:18:23,240 --> 00:18:25,090
to get that far.

375
00:18:25,090 --> 00:18:28,195
So let's just see
how big that is.

376
00:18:28,195 --> 00:18:30,300
All right.

377
00:18:30,300 --> 00:18:31,760
I think I worked this out.

378
00:18:31,760 --> 00:18:42,010

379
00:18:42,010 --> 00:18:43,170
So let's see.

380
00:18:43,170 --> 00:18:48,020
That means that N is equal
to e to the 24th--

381
00:18:48,020 --> 00:18:53,030
and if you realize that
these blocks are

382
00:18:53,030 --> 00:18:54,400
3 centimeters high--

383
00:18:54,400 --> 00:18:57,160

384
00:18:57,160 --> 00:19:00,450
OK let's see how many that
we would need here.

385
00:19:00,450 --> 00:19:02,210
That's kind of a lot.

386
00:19:02,210 --> 00:19:09,800
Let's see, it's 3 centimeters
times e to the 24th, which is

387
00:19:09,800 --> 00:19:15,660
about 8 times 10 to
the 8th meters.

388
00:19:15,660 --> 00:19:17,600
OK.

389
00:19:17,600 --> 00:19:20,810
And that is twice the distance
to the moon.

390
00:19:20,810 --> 00:19:32,310

391
00:19:32,310 --> 00:19:35,190
So OK, so I could do it maybe.

392
00:19:35,190 --> 00:19:37,980
But I would need a
lot of blocks.

393
00:19:37,980 --> 00:19:38,435
Right?

394
00:19:38,435 --> 00:19:42,190
So that's not very plausible
here, all right.

395
00:19:42,190 --> 00:19:45,500
So those of you who voted
against this were actually

396
00:19:45,500 --> 00:19:47,080
sort of half right.

397
00:19:47,080 --> 00:19:50,340
And in fact, if you wanted to
get it to the wall over there,

398
00:19:50,340 --> 00:19:54,730
which is over 30 feet, the
height would be about the

399
00:19:54,730 --> 00:19:57,360
diameter of the observable
universe.

400
00:19:57,360 --> 00:20:01,580
That's kind of a long way.

401
00:20:01,580 --> 00:20:06,440
There's one other thing that I
wanted to point out to you

402
00:20:06,440 --> 00:20:09,340
about this shape here.

403
00:20:09,340 --> 00:20:13,610
Which is that if you lean
to the left, right?

404
00:20:13,610 --> 00:20:15,460
If you put your head like this--
of course you have to

405
00:20:15,460 --> 00:20:23,920
be on your side to look at it--
this curve is the shape

406
00:20:23,920 --> 00:20:25,810
of a logarithmic curve.

407
00:20:25,810 --> 00:20:28,360
So in other words, if you think
of the vertical as the

408
00:20:28,360 --> 00:20:32,115
x-axis, and the horizontal that
way, is the vertical, is

409
00:20:32,115 --> 00:20:36,640
the up direction, then this
thing is growing very, very,

410
00:20:36,640 --> 00:20:38,510
very, very slowly.

411
00:20:38,510 --> 00:20:43,460
If you send the x-axis all the
way up to the moon, the graph

412
00:20:43,460 --> 00:20:47,540
still hasn't gotten across
the lab tables here.

413
00:20:47,540 --> 00:20:48,905
It's only partway there.

414
00:20:48,905 --> 00:20:52,290
If you go twice the distance to
the moon up that way, it's

415
00:20:52,290 --> 00:20:54,340
gotten finally to that end.

416
00:20:54,340 --> 00:20:56,990
All right so that's how slowly
the logarithm grows.

417
00:20:56,990 --> 00:20:58,650
It grows very, very slowly.

418
00:20:58,650 --> 00:21:00,460
And if you look at it another
way, if you stand on your

419
00:21:00,460 --> 00:21:05,420
head, you can see an
exponential curve.

420
00:21:05,420 --> 00:21:07,950
So you get some sense
as to the growth

421
00:21:07,950 --> 00:21:10,420
properties of these functions.

422
00:21:10,420 --> 00:21:17,120
And fortunately these are
protecting us from all kinds

423
00:21:17,120 --> 00:21:20,200
of stuff that would happen if
there weren't exponentially

424
00:21:20,200 --> 00:21:21,750
small tails in the world.

425
00:21:21,750 --> 00:21:23,960
Like you know, I could walk
through this wall which I

426
00:21:23,960 --> 00:21:27,000
wouldn't like doing.

427
00:21:27,000 --> 00:21:32,410
OK, now so this is
our last example.

428
00:21:32,410 --> 00:21:35,100
And the important number,
unfortunately we didn't

429
00:21:35,100 --> 00:21:36,790
discover another important
number.

430
00:21:36,790 --> 00:21:40,250
There wasn't an amazing number
place where this stopped.

431
00:21:40,250 --> 00:21:44,160
All we discovered again is some
property of infinity.

432
00:21:44,160 --> 00:21:46,150
So infinity is still
a nice number.

433
00:21:46,150 --> 00:21:50,120
And the theme here is just that
infinity isn't just one

434
00:21:50,120 --> 00:21:54,460
thing, it has a character which
is a rate of growth.

435
00:21:54,460 --> 00:21:55,840
And you shouldn't just
think of there being

436
00:21:55,840 --> 00:21:57,330
one order of infinity.

437
00:21:57,330 --> 00:21:58,530
There are lots of different
orders.

438
00:21:58,530 --> 00:22:01,280
And some of them have different
meaning from others.

439
00:22:01,280 --> 00:22:05,590
All right so that's the theme I
wanted to do, and just have

440
00:22:05,590 --> 00:22:07,900
a visceral example
of infinity.

441
00:22:07,900 --> 00:22:15,110
Now, we're going to move on
now to some other kinds of

442
00:22:15,110 --> 00:22:16,210
techniques.

443
00:22:16,210 --> 00:22:20,790
And this is going to be
our last subject.

444
00:22:20,790 --> 00:22:24,020
What we're going to talk
about is what are

445
00:22:24,020 --> 00:22:27,320
known as power series.

446
00:22:27,320 --> 00:22:29,920
And we've already seen our
first power series.

447
00:22:29,920 --> 00:22:32,710

448
00:22:32,710 --> 00:22:35,240
And I'm going to remind
you of that.

449
00:22:35,240 --> 00:22:43,980

450
00:22:43,980 --> 00:22:45,490
Here we are with power series.

451
00:22:45,490 --> 00:22:51,490

452
00:22:51,490 --> 00:22:53,980
Our first series was this one.

453
00:22:53,980 --> 00:22:58,870

454
00:22:58,870 --> 00:23:02,890
And we mentioned last time that
it was equal to 1 over 1

455
00:23:02,890 --> 00:23:06,430
minus x, for x less than 1.

456
00:23:06,430 --> 00:23:09,770

457
00:23:09,770 --> 00:23:12,040
Well this one is known as
the geometric series.

458
00:23:12,040 --> 00:23:13,830
You didn't use the letter
x last time, I

459
00:23:13,830 --> 00:23:15,050
used the letter a.

460
00:23:15,050 --> 00:23:16,910
But this is known as the
geometric series.

461
00:23:16,910 --> 00:23:24,950

462
00:23:24,950 --> 00:23:31,800
Now I'm going to show you one
reason why this is true, why

463
00:23:31,800 --> 00:23:33,900
the formula holds.

464
00:23:33,900 --> 00:23:37,680
And it's just the kind of
manipulation that was done

465
00:23:37,680 --> 00:23:41,270
when these things were
first introduced.

466
00:23:41,270 --> 00:23:46,250
And here's the idea
of a proof.

467
00:23:46,250 --> 00:23:53,590
So suppose that this sum is
equal to some number S, which

468
00:23:53,590 --> 00:23:57,360
is the sum of all of
these numbers here.

469
00:23:57,360 --> 00:24:00,540

470
00:24:00,540 --> 00:24:02,570
The first thing that I'm going
to do is I'm going

471
00:24:02,570 --> 00:24:05,500
to multiply by x.

472
00:24:05,500 --> 00:24:08,326
OK, so if I multiply by x--

473
00:24:08,326 --> 00:24:09,560
let's think about that--

474
00:24:09,560 --> 00:24:13,810
I multiply by x on both the left
and the right-hand side.

475
00:24:13,810 --> 00:24:19,595
Then on the left side, I get x
plus x-squared plus x-cubed

476
00:24:19,595 --> 00:24:20,990
plus, and so forth.

477
00:24:20,990 --> 00:24:23,180
And on the right side,
I get S times x.

478
00:24:23,180 --> 00:24:27,380

479
00:24:27,380 --> 00:24:31,150
And now I'm going to subtract
the two equations,

480
00:24:31,150 --> 00:24:32,910
one from the other.

481
00:24:32,910 --> 00:24:36,230
And there's a very, very
substantial cancellation.

482
00:24:36,230 --> 00:24:39,180
This whole tail here
gets canceled off.

483
00:24:39,180 --> 00:24:41,030
And the only thing that's
left is the 1.

484
00:24:41,030 --> 00:24:45,950
So when I subtract, I get
1 on the left-hand side.

485
00:24:45,950 --> 00:24:53,090
And on the right-hand side,
I get S minus S times x.

486
00:24:53,090 --> 00:24:54,340
All right?

487
00:24:54,340 --> 00:24:58,900

488
00:24:58,900 --> 00:25:03,850
And now that can be rewritten
as S times 1 minus x.

489
00:25:03,850 --> 00:25:06,180
And so I've got my
formula here.

490
00:25:06,180 --> 00:25:13,553
This is 1 over 1 minus x is
equal to S. All right.

491
00:25:13,553 --> 00:25:17,850

492
00:25:17,850 --> 00:25:26,340
Now this reasoning
has one flaw.

493
00:25:26,340 --> 00:25:28,120
It's not complete.

494
00:25:28,120 --> 00:25:33,080
And this reasoning is
basically correct.

495
00:25:33,080 --> 00:25:45,680
But it's incomplete because
it requires that S exists.

496
00:25:45,680 --> 00:25:50,270

497
00:25:50,270 --> 00:25:53,050
For example, it doesn't
make any sense in the

498
00:25:53,050 --> 00:25:55,100
case x equals 1.

499
00:25:55,100 --> 00:25:59,400
So for example in the case x
equals 1, we have 1 plus 1

500
00:25:59,400 --> 00:26:04,060
plus 1 plus et cetera, equals
whatever we call S. And then

501
00:26:04,060 --> 00:26:08,798
when we multiply through by 1,
we get 1 plus 1 plus 1 plus,

502
00:26:08,798 --> 00:26:10,640
equals S times 1.

503
00:26:10,640 --> 00:26:13,060
And now you see that the
subtraction gives us infinity

504
00:26:13,060 --> 00:26:16,330
minus infinity is equal to
infinity minus infinity.

505
00:26:16,330 --> 00:26:17,580
That's what's really
going on in the

506
00:26:17,580 --> 00:26:19,790
argument in this context.

507
00:26:19,790 --> 00:26:21,340
So it's just nonsense.

508
00:26:21,340 --> 00:26:24,380
I mean it doesn't give us
anything meaningful.

509
00:26:24,380 --> 00:26:27,090
So this argument, it's great.

510
00:26:27,090 --> 00:26:30,000
And it gives us the right
answer, but not always.

511
00:26:30,000 --> 00:26:32,310
And the times when it gives
us the answer, the correct

512
00:26:32,310 --> 00:26:36,800
answer, is when the series
is convergent.

513
00:26:36,800 --> 00:26:38,800
And that's why we care
about convergence.

514
00:26:38,800 --> 00:26:42,060
Because we want manipulations
like this to be allowed.

515
00:26:42,060 --> 00:26:47,780

516
00:26:47,780 --> 00:26:50,620
So the good case, this is the
red case that we were

517
00:26:50,620 --> 00:26:54,790
describing last time,
that's the bad case.

518
00:26:54,790 --> 00:26:58,820
But what we want is the good
case, the convergent case.

519
00:26:58,820 --> 00:27:01,770
And that is the case when
x is less than 1.

520
00:27:01,770 --> 00:27:03,182
So this is the convergent
case.

521
00:27:03,182 --> 00:27:11,290

522
00:27:11,290 --> 00:27:11,960
Yep.

523
00:27:11,960 --> 00:27:14,640
OK, so they're much more
detailed things to check

524
00:27:14,640 --> 00:27:15,710
exactly what's going on.

525
00:27:15,710 --> 00:27:18,240
But I'm going to just say
general words about how you

526
00:27:18,240 --> 00:27:20,160
recognize convergence.

527
00:27:20,160 --> 00:27:22,370
And then we're not going to
worry about so much about

528
00:27:22,370 --> 00:27:24,740
convergence, because it
works very, very well.

529
00:27:24,740 --> 00:27:27,700
And it's always easy to
diagnose when there's

530
00:27:27,700 --> 00:27:31,590
convergence with
a power series.

531
00:27:31,590 --> 00:27:33,450
All right so here's
the general setup.

532
00:27:33,450 --> 00:27:44,700

533
00:27:44,700 --> 00:27:50,130
The general setup is that we
have not just the coefficients

534
00:27:50,130 --> 00:27:56,660
1 all the time, but any numbers
here, dot, dot, dot.

535
00:27:56,660 --> 00:27:59,310
And we abbreviate that with
the summation notation.

536
00:27:59,310 --> 00:28:05,360
This is the sum a n x to the
n, n equals 0 to infinity.

537
00:28:05,360 --> 00:28:07,260
And that's what's known
as a power series.

538
00:28:07,260 --> 00:28:12,370

539
00:28:12,370 --> 00:28:18,380
Fortunately there is a very
simple rule about how power

540
00:28:18,380 --> 00:28:20,260
series converge.

541
00:28:20,260 --> 00:28:23,200
And it's the following.

542
00:28:23,200 --> 00:28:28,480
There's a magic number R which
depends on these numbers here

543
00:28:28,480 --> 00:28:29,340
such that--

544
00:28:29,340 --> 00:28:31,460
and this thing is known as
a radius of convergence--

545
00:28:31,460 --> 00:28:37,020

546
00:28:37,020 --> 00:28:40,090
and the problem that we had,
it's this number 1 here.

547
00:28:40,090 --> 00:28:43,030
This thing works for
x less than 1.

548
00:28:43,030 --> 00:28:47,060
In our case, it's maybe x less
R. So that's some symmetric

549
00:28:47,060 --> 00:28:47,960
interval, right?

550
00:28:47,960 --> 00:28:55,390
That's the same as minus R, less
then x, less than R, and

551
00:28:55,390 --> 00:28:58,310
so where there's convergence.

552
00:28:58,310 --> 00:29:00,575
OK, where the series converges.

553
00:29:00,575 --> 00:29:01,825
Converges.

554
00:29:01,825 --> 00:29:07,210

555
00:29:07,210 --> 00:29:13,270
And then there's the region
where every computation that

556
00:29:13,270 --> 00:29:16,500
you give will give
you nonsense.

557
00:29:16,500 --> 00:29:24,740
So x greater than R is the sum
a sub n x to the n, diverges.

558
00:29:24,740 --> 00:29:28,710

559
00:29:28,710 --> 00:29:43,780
And x equals R is very delicate,
borderline, and will

560
00:29:43,780 --> 00:29:45,030
not be used by us.

561
00:29:45,030 --> 00:29:50,060

562
00:29:50,060 --> 00:29:52,660
OK, we're going to stick
inside the radius of

563
00:29:52,660 --> 00:29:54,750
convergence.

564
00:29:54,750 --> 00:29:56,610
Now the way you'll be
able to recognize

565
00:29:56,610 --> 00:29:58,920
this, is the following.

566
00:29:58,920 --> 00:30:05,370
What always happens is that
these numbers tend to 0

567
00:30:05,370 --> 00:30:19,500
exponentially fast, fast for x
in R, and doesn't even tend to

568
00:30:19,500 --> 00:30:28,470
0 at all for x greater
than R. All right so

569
00:30:28,470 --> 00:30:30,200
it'll be totally obvious.

570
00:30:30,200 --> 00:30:33,495
When you look at this series
here, what's happening when x

571
00:30:33,495 --> 00:30:35,550
less than R is that the numbers
are getting smaller

572
00:30:35,550 --> 00:30:37,990
and smaller, less than 1.

573
00:30:37,990 --> 00:30:39,350
When x is bigger than
1, the numbers are

574
00:30:39,350 --> 00:30:40,100
getting bigger and bigger.

575
00:30:40,100 --> 00:30:42,850
There's no chance that
the series converges.

576
00:30:42,850 --> 00:30:45,870
So that's going to be the case
with all power series.

577
00:30:45,870 --> 00:30:47,680
There's going to be a cut off.

578
00:30:47,680 --> 00:30:49,810
And it'll be one particular
number.

579
00:30:49,810 --> 00:30:51,530
And below that it'll be
obvious that you have

580
00:30:51,530 --> 00:30:53,260
convergence, and you'll be
able to do computations.

581
00:30:53,260 --> 00:30:55,690
And above that every formula
will be wrong

582
00:30:55,690 --> 00:30:57,220
and won't make sense.

583
00:30:57,220 --> 00:30:59,370
So it's a very clean thing.

584
00:30:59,370 --> 00:31:02,210
There is this very subtle
borderline, but we're not

585
00:31:02,210 --> 00:31:04,350
going to discuss that
in this class.

586
00:31:04,350 --> 00:31:07,910
And it's actually not
used in direct

587
00:31:07,910 --> 00:31:10,383
studies of power series.

588
00:31:10,383 --> 00:31:13,820
AUDIENCE: How can you tell when
the numbers are declining

589
00:31:13,820 --> 00:31:17,440
exponentially fast, whereas
just, in other words 1 over x

590
00:31:17,440 --> 00:31:18,030
[INAUDIBLE]?

591
00:31:18,030 --> 00:31:22,090
PROFESSOR: OK so, the question
is why was I able to tell you

592
00:31:22,090 --> 00:31:23,070
this word here?

593
00:31:23,070 --> 00:31:25,720
Why was I able to tell you not
only is it going to 0, but

594
00:31:25,720 --> 00:31:27,550
it's going exponentially fast?

595
00:31:27,550 --> 00:31:29,000
I'm telling you extra
information.

596
00:31:29,000 --> 00:31:32,830
I'm telling you it always goes
exponentially fast. You can

597
00:31:32,830 --> 00:31:34,080
identify it.

598
00:31:34,080 --> 00:31:36,150

599
00:31:36,150 --> 00:31:37,620
In other words, you'll see it.

600
00:31:37,620 --> 00:31:39,490
And it will happen every
single time.

601
00:31:39,490 --> 00:31:41,600
I'm just promising you that
it works that way.

602
00:31:41,600 --> 00:31:44,830
And it's really for the same
reason that it works that way

603
00:31:44,830 --> 00:31:46,840
here, that these are powers.

604
00:31:46,840 --> 00:31:49,430
And what's going on over here
is there are, it's close to

605
00:31:49,430 --> 00:31:52,181
powers with this a n's.

606
00:31:52,181 --> 00:31:54,640
All right?

607
00:31:54,640 --> 00:31:56,410
There's a long discussion
of radius of

608
00:31:56,410 --> 00:31:58,350
convergence in many textbooks.

609
00:31:58,350 --> 00:32:04,670
But really it's not necessary,
all right, for this purpose?

610
00:32:04,670 --> 00:32:05,418
Yeah?

611
00:32:05,418 --> 00:32:07,090
AUDIENCE: How do you find R?

612
00:32:07,090 --> 00:32:08,400
PROFESSOR: The question
was how do you find R?

613
00:32:08,400 --> 00:32:10,580
Yes, so I just said, there's a
long discussion for how you

614
00:32:10,580 --> 00:32:12,660
find the radius of convergence
in textbooks.

615
00:32:12,660 --> 00:32:15,800
But we will not be discussing
that here.

616
00:32:15,800 --> 00:32:17,680
And it won't be necessary
for you.

617
00:32:17,680 --> 00:32:19,430
Because it will be obvious
in any given

618
00:32:19,430 --> 00:32:20,890
series what the R is.

619
00:32:20,890 --> 00:32:23,650
It will always either
1 or infinity.

620
00:32:23,650 --> 00:32:25,850
It will always work for
all x, or maybe it'll

621
00:32:25,850 --> 00:32:26,820
stop at some point.

622
00:32:26,820 --> 00:32:31,270
But it'll be very clear where
it stops, as it is for the

623
00:32:31,270 --> 00:32:33,201
geometric series.

624
00:32:33,201 --> 00:32:36,070
All right?

625
00:32:36,070 --> 00:32:40,170
OK, so now I need to give you
the basic facts, and give you

626
00:32:40,170 --> 00:32:41,420
a few examples.

627
00:32:41,420 --> 00:32:44,400

628
00:32:44,400 --> 00:32:46,420
So why are we looking
at these series?

629
00:32:46,420 --> 00:32:51,240

630
00:32:51,240 --> 00:32:55,010
Well the answer is we're
looking at these series

631
00:32:55,010 --> 00:33:00,060
because the role that they play
is exactly the reverse of

632
00:33:00,060 --> 00:33:03,420
this equation here.

633
00:33:03,420 --> 00:33:06,030
That is, and this is a theme
which I have tried to

634
00:33:06,030 --> 00:33:08,640
emphasize throughout this
course, you can read

635
00:33:08,640 --> 00:33:11,870
equalities in two directions.

636
00:33:11,870 --> 00:33:15,610
Both are interesting,
typically.

637
00:33:15,610 --> 00:33:18,220
You can think, I don't know
what the value of this is.

638
00:33:18,220 --> 00:33:19,240
Here's a way of evaluating.

639
00:33:19,240 --> 00:33:22,060
And in other words, the right
side is a formula

640
00:33:22,060 --> 00:33:23,090
for the left side.

641
00:33:23,090 --> 00:33:26,350
Or you can think of the left
side as being a formula for

642
00:33:26,350 --> 00:33:27,600
the right side.

643
00:33:27,600 --> 00:33:30,720

644
00:33:30,720 --> 00:33:34,350
And the idea of series is that
they're flexible enough to

645
00:33:34,350 --> 00:33:36,060
represent all of the functions
that we've

646
00:33:36,060 --> 00:33:39,300
encountered in this course.

647
00:33:39,300 --> 00:33:41,780
This is the tool which is very
much like the decimal

648
00:33:41,780 --> 00:33:44,013
expansion which allows you to
represent numbers like the

649
00:33:44,013 --> 00:33:45,130
square root of 2.

650
00:33:45,130 --> 00:33:47,520
Now we're going to be
representing all the numbers,

651
00:33:47,520 --> 00:33:51,460
all the functions that we know,
e to the x, arctangent,

652
00:33:51,460 --> 00:33:52,550
sine, cosine.

653
00:33:52,550 --> 00:33:55,440
All of those functions become
completely flexible, and

654
00:33:55,440 --> 00:33:57,970
completely available to us, and
computationally available

655
00:33:57,970 --> 00:33:59,900
to us directly.

656
00:33:59,900 --> 00:34:01,440
So that's what this
is a tool for.

657
00:34:01,440 --> 00:34:04,250
And it's just like decimal
expansions giving you handle

658
00:34:04,250 --> 00:34:05,500
on all real numbers.

659
00:34:05,500 --> 00:34:08,920

660
00:34:08,920 --> 00:34:12,110
So here's how it works.

661
00:34:12,110 --> 00:34:33,600
The rules for convergent power
series are just like

662
00:34:33,600 --> 00:34:34,850
polynomials.

663
00:34:34,850 --> 00:34:41,280

664
00:34:41,280 --> 00:34:43,960
All of the manipulations that
you do for power series are

665
00:34:43,960 --> 00:34:46,120
essentially the same
as for polynomials.

666
00:34:46,120 --> 00:34:49,190
So what kinds of things do
we do with polynomials?

667
00:34:49,190 --> 00:34:50,440
We add them.

668
00:34:50,440 --> 00:34:53,280

669
00:34:53,280 --> 00:34:54,530
We multiply them together.

670
00:34:54,530 --> 00:34:57,630

671
00:34:57,630 --> 00:35:01,060
We do substitutions.

672
00:35:01,060 --> 00:35:01,290
Right?

673
00:35:01,290 --> 00:35:04,970
We take one function of
another function.

674
00:35:04,970 --> 00:35:06,220
We divide them.

675
00:35:06,220 --> 00:35:10,110

676
00:35:10,110 --> 00:35:11,590
OK.

677
00:35:11,590 --> 00:35:15,040
And these are all really not
very surprising operations.

678
00:35:15,040 --> 00:35:17,710
And we will be able to do them
with power series too.

679
00:35:17,710 --> 00:35:20,500
The ones that are interesting,
really interesting for

680
00:35:20,500 --> 00:35:24,360
calculus, are the last two.

681
00:35:24,360 --> 00:35:29,790
We differentiate them, and
we integrate them.

682
00:35:29,790 --> 00:35:34,070

683
00:35:34,070 --> 00:35:37,120
And all of these operations
we'll be able to do for power

684
00:35:37,120 --> 00:35:38,370
series as well.

685
00:35:38,370 --> 00:35:42,950

686
00:35:42,950 --> 00:35:49,120
So now let's explain the
high points of this.

687
00:35:49,120 --> 00:35:51,750
Which is mainly just the
differentiation and the

688
00:35:51,750 --> 00:35:53,110
integration part.

689
00:35:53,110 --> 00:36:03,850
So if I take a series like this
and so forth, the formula

690
00:36:03,850 --> 00:36:08,290
for it's derivative is just
like polynomials.

691
00:36:08,290 --> 00:36:10,220
That's what I just said, it's
just like polynomials.

692
00:36:10,220 --> 00:36:12,570
So the derivative of
the constant is 0.

693
00:36:12,570 --> 00:36:15,850
The derivative of this
term is a 1.

694
00:36:15,850 --> 00:36:19,340
This one is plus 2 a 2 x.

695
00:36:19,340 --> 00:36:23,920
This one is 3a 3 x-squared,
et cetera.

696
00:36:23,920 --> 00:36:27,070
That's the formula.

697
00:36:27,070 --> 00:36:41,820
Similarly if I integrate, well
there's an unknown constant

698
00:36:41,820 --> 00:36:45,250
which I'm going to put first
rather than last. Which

699
00:36:45,250 --> 00:36:47,200
corresponds sort of to the
a 0 term which, is going

700
00:36:47,200 --> 00:36:48,010
to get wiped out.

701
00:36:48,010 --> 00:36:50,770
That a 0 term suddenly
becomes a 0 x.

702
00:36:50,770 --> 00:36:55,210
And the anti-derivative of
this next term is a sub 1

703
00:36:55,210 --> 00:36:56,980
x-squared over 2.

704
00:36:56,980 --> 00:37:02,680
And the next term is a 2 x-cubed
over 3, and so forth.

705
00:37:02,680 --> 00:37:05,200

706
00:37:05,200 --> 00:37:06,456
Yeah, question?

707
00:37:06,456 --> 00:37:08,690
AUDIENCE: Is that a series
or a polynomial?

708
00:37:08,690 --> 00:37:10,190
PROFESSOR: Is this a series
or a polynomial?

709
00:37:10,190 --> 00:37:11,180
Good question.

710
00:37:11,180 --> 00:37:14,700
It's a polynomial if it ends.

711
00:37:14,700 --> 00:37:19,490
If it goes on infinitely far,
then it's a series.

712
00:37:19,490 --> 00:37:22,420
They look practically the same,
polynomials and series.

713
00:37:22,420 --> 00:37:26,400
There's this little dot,
dot, dot here.

714
00:37:26,400 --> 00:37:27,660
Is this a series or
a polynomial?

715
00:37:27,660 --> 00:37:28,780
It's the same rule.

716
00:37:28,780 --> 00:37:30,460
If it stops at a finite
stage, this one

717
00:37:30,460 --> 00:37:32,210
stops at a finite stage.

718
00:37:32,210 --> 00:37:35,074
If it goes on forever,
it goes on forever.

719
00:37:35,074 --> 00:37:37,022
AUDIENCE: So I thought
that the series

720
00:37:37,022 --> 00:37:37,980
add up finite numbers.

721
00:37:37,980 --> 00:37:41,735
You can add up terms
of x in series?

722
00:37:41,735 --> 00:37:45,520

723
00:37:45,520 --> 00:37:47,380
PROFESSOR: So an interesting
question.

724
00:37:47,380 --> 00:37:51,350
So the question that was just
asked is I thought that a

725
00:37:51,350 --> 00:37:53,060
series added up finite
numbers.

726
00:37:53,060 --> 00:37:55,140
You could add up x?

727
00:37:55,140 --> 00:37:56,900
That was what you said, right?

728
00:37:56,900 --> 00:38:02,560
OK now notice that I pulled that
off on you by changing

729
00:38:02,560 --> 00:38:07,350
the letter a to the letter x at
the very beginning of this

730
00:38:07,350 --> 00:38:10,170
commentary here.

731
00:38:10,170 --> 00:38:11,760
This is a series.

732
00:38:11,760 --> 00:38:15,630
For each individual value
of x, it's a number.

733
00:38:15,630 --> 00:38:17,848
So in other words, it I plug
in here x equals 1/2, I'm

734
00:38:17,848 --> 00:38:21,090
going to add 1 plus 1/2 plus 1/4
plus 1/8; and I'll get a

735
00:38:21,090 --> 00:38:22,470
number which is 2.

736
00:38:22,470 --> 00:38:25,030
And I'll plug in a number over
here, and I'll get a number.

737
00:38:25,030 --> 00:38:27,820
On the other hand, I can do
this for each value of x.

738
00:38:27,820 --> 00:38:31,770
So the interpretation of this is
that it's a function of x.

739
00:38:31,770 --> 00:38:34,380
And similarly this is
a function of x.

740
00:38:34,380 --> 00:38:38,890
It works when you plug in the
possible values x between

741
00:38:38,890 --> 00:38:42,330
minus 1 and 1.

742
00:38:42,330 --> 00:38:45,060
So there's really no distinction
there, it's just I

743
00:38:45,060 --> 00:38:47,010
slipped it passed you.

744
00:38:47,010 --> 00:38:48,260
These are functions of x.

745
00:38:48,260 --> 00:38:50,790

746
00:38:50,790 --> 00:38:53,290
And the notion of a power series
is this idea that you

747
00:38:53,290 --> 00:38:55,860
put coefficients on a series,
but then you allow yourself

748
00:38:55,860 --> 00:38:58,990
the flexibility to stick
powers here.

749
00:38:58,990 --> 00:39:02,120
And that's exactly
what we're doing.

750
00:39:02,120 --> 00:39:04,120
OK there are other kinds of
series where you stick other

751
00:39:04,120 --> 00:39:06,120
interesting functions in here
like sines and cosines.

752
00:39:06,120 --> 00:39:08,620
There are lots of other series
that people study.

753
00:39:08,620 --> 00:39:10,465
And these are the
simplest ones.

754
00:39:10,465 --> 00:39:13,370
And all those examples are
extremely helpful for

755
00:39:13,370 --> 00:39:14,515
representing functions.

756
00:39:14,515 --> 00:39:18,920
But we're only going to
do this example here.

757
00:39:18,920 --> 00:39:23,610
All right, so here are
the two rules.

758
00:39:23,610 --> 00:39:30,520
And now there's only one other
complication here which I have

759
00:39:30,520 --> 00:39:36,180
to explain to you before giving
you a bunch of examples

760
00:39:36,180 --> 00:39:41,270
to show you that this works
extremely well.

761
00:39:41,270 --> 00:39:45,090
And the last thing that I have
to do for you is explain to

762
00:39:45,090 --> 00:39:46,580
you something called
Taylor's formula.

763
00:39:46,580 --> 00:39:55,280

764
00:39:55,280 --> 00:39:59,930
Taylor's formula is the
way you get from the

765
00:39:59,930 --> 00:40:02,970
representations that we're
used to a functions, to a

766
00:40:02,970 --> 00:40:06,180
representation in the form
of these coefficients.

767
00:40:06,180 --> 00:40:09,090
When I gave you the function e
to the x, it didn't look like

768
00:40:09,090 --> 00:40:11,280
a polynomial.

769
00:40:11,280 --> 00:40:14,940
And we have to figure out which
of these guys it is, if

770
00:40:14,940 --> 00:40:19,090
it's going to fall into
our category here.

771
00:40:19,090 --> 00:40:20,370
And here's the formula.

772
00:40:20,370 --> 00:40:23,430
I'll explain to you how
it works in a second.

773
00:40:23,430 --> 00:40:26,390
So the formula is f of x, turns
out there's a formula in

774
00:40:26,390 --> 00:40:28,560
terms of the derivatives of f.

775
00:40:28,560 --> 00:40:32,180
Namely, you differentiate n
times, and you evaluate it at

776
00:40:32,180 --> 00:40:34,670
0, and you divide by
n factorial, and

777
00:40:34,670 --> 00:40:37,640
multiply by x to the n.

778
00:40:37,640 --> 00:40:40,510
So here's Taylor's formula.

779
00:40:40,510 --> 00:40:44,270
This tells you what the
Taylor series is.

780
00:40:44,270 --> 00:40:48,720
Now about half of our job for
the next few minutes is going

781
00:40:48,720 --> 00:40:51,630
to be to give examples
of this.

782
00:40:51,630 --> 00:40:56,810
But let me just explain to
you why this has to be.

783
00:40:56,810 --> 00:41:00,870
If you pick out this number
here, this is the a n, the

784
00:41:00,870 --> 00:41:03,290
magic number a n here.

785
00:41:03,290 --> 00:41:05,450
So let's just illustrate it.

786
00:41:05,450 --> 00:41:12,610
If f of x happens to be a zero
plus a 1 x plus a 2 x-squared

787
00:41:12,610 --> 00:41:15,940
plus a 3 x-cubed plus
dot, dot, dot.

788
00:41:15,940 --> 00:41:19,810
And now I differentiate
it, right?

789
00:41:19,810 --> 00:41:26,930
I get a 1 plus 2 a
2x plus 3 a 3 x.

790
00:41:26,930 --> 00:41:29,510
If I differentiate it
another time, I

791
00:41:29,510 --> 00:41:32,990
get 2 a 2, plus 3--sorry--

792
00:41:32,990 --> 00:41:38,560
3 times 2 a 3 x plus
dot, dot, dot.

793
00:41:38,560 --> 00:41:47,360
And now a third time, I get 3
times 2 a 3 plus et cetera.

794
00:41:47,360 --> 00:41:50,000
So this next term is really
in disguise, 4

795
00:41:50,000 --> 00:41:53,470
times 3 times 2 x--

796
00:41:53,470 --> 00:41:56,410
a sorry-- a 4x.

797
00:41:56,410 --> 00:41:59,070
That's what really comes down if
I kept track of the fourth

798
00:41:59,070 --> 00:42:00,980
term there.

799
00:42:00,980 --> 00:42:04,660
So now here is my function.

800
00:42:04,660 --> 00:42:11,110
But now you see if I plug in x
equals 0, I can pick off the

801
00:42:11,110 --> 00:42:13,750
third term.

802
00:42:13,750 --> 00:42:21,590
f triple prime of 0 is equal
to 3 times 2 times a 3.

803
00:42:21,590 --> 00:42:23,620
Right, because all the rest of
those terms when I plug in 0

804
00:42:23,620 --> 00:42:25,080
are just 0.

805
00:42:25,080 --> 00:42:26,730
Here's the formula.

806
00:42:26,730 --> 00:42:30,610
And so the pattern
here is this.

807
00:42:30,610 --> 00:42:32,860
And what's really going on here
is this is really 3 times

808
00:42:32,860 --> 00:42:37,110
2 times 1, a 3.

809
00:42:37,110 --> 00:42:48,480
And in general a n is equal
to f nth derivative

810
00:42:48,480 --> 00:42:50,410
divided by n factorial.

811
00:42:50,410 --> 00:42:53,990
And of course, n factorial, I
remind you, is n times n minus

812
00:42:53,990 --> 00:42:57,570
1 times n minus 2, all
the way down to 1.

813
00:42:57,570 --> 00:43:02,600

814
00:43:02,600 --> 00:43:05,980
Now there's one more
crazy convention

815
00:43:05,980 --> 00:43:08,220
which is always used.

816
00:43:08,220 --> 00:43:10,960
Which is that there's something
very strange here

817
00:43:10,960 --> 00:43:15,210
down at 0, which is that 0
factorials turns out, has to

818
00:43:15,210 --> 00:43:16,990
be set equal to 1.

819
00:43:16,990 --> 00:43:19,730
All right, so that's what you
do in order to make this

820
00:43:19,730 --> 00:43:20,540
formula work out.

821
00:43:20,540 --> 00:43:22,425
And that's one of the reasons
for this convention.

822
00:43:22,425 --> 00:43:29,470

823
00:43:29,470 --> 00:43:30,600
All right.

824
00:43:30,600 --> 00:43:36,680
So my next goal is to give
you some examples.

825
00:43:36,680 --> 00:43:45,280
And let's do a couple.

826
00:43:45,280 --> 00:43:48,130

827
00:43:48,130 --> 00:43:54,050
So here's, well you know, I'm
going to have to let you see a

828
00:43:54,050 --> 00:43:56,120
few of them next time.

829
00:43:56,120 --> 00:44:00,500
But let me just tell you this
one, which is by far the most

830
00:44:00,500 --> 00:44:01,750
impressive.

831
00:44:01,750 --> 00:44:06,370

832
00:44:06,370 --> 00:44:10,580
So what happens with e to the
x, if the function of f of x

833
00:44:10,580 --> 00:44:18,190
is e to the x, is that it's
derivative is also e to the x.

834
00:44:18,190 --> 00:44:21,390
And its second derivative
is also e to the x.

835
00:44:21,390 --> 00:44:23,010
And it just keeps on
going that way.

836
00:44:23,010 --> 00:44:25,180
They're all the same.

837
00:44:25,180 --> 00:44:30,010
So that means that these numbers
in Taylor's formula,

838
00:44:30,010 --> 00:44:33,130
in the numerator, the
nth derivative is

839
00:44:33,130 --> 00:44:37,090
very easy to evaluate.

840
00:44:37,090 --> 00:44:39,280
It's just e to the x.

841
00:44:39,280 --> 00:44:43,890
And if I evaluated at x equals
0, I just get 1.

842
00:44:43,890 --> 00:44:46,990
So all of those numerators
are 1.

843
00:44:46,990 --> 00:44:51,850
So the formula here, is the sum
n equals 0 to infinity, of

844
00:44:51,850 --> 00:44:55,987
1 divided by n factorial
x to the n.

845
00:44:55,987 --> 00:45:00,960

846
00:45:00,960 --> 00:45:04,890
In particular, we now have an
honest formula for e to the

847
00:45:04,890 --> 00:45:06,800
first power.

848
00:45:06,800 --> 00:45:08,330
Which is just e.

849
00:45:08,330 --> 00:45:11,970
Which if I plug it in,
x equals 1, I get 1.

850
00:45:11,970 --> 00:45:14,750
This is the n equals
0 term plus 1.

851
00:45:14,750 --> 00:45:19,440
This is the n equals 1 term plus
1 over 2 factorial plus 1

852
00:45:19,440 --> 00:45:21,991
over 3 factorial plus
1 over 4 factorial.

853
00:45:21,991 --> 00:45:27,070

854
00:45:27,070 --> 00:45:27,540
Right?

855
00:45:27,540 --> 00:45:31,360
So this is our first honest
formula for e.

856
00:45:31,360 --> 00:45:33,960
And also, this is
how you compute

857
00:45:33,960 --> 00:45:35,340
the exponential function.

858
00:45:35,340 --> 00:45:41,750

859
00:45:41,750 --> 00:45:49,980
Finally if you take a function
like sine x, what you'll

860
00:45:49,980 --> 00:45:52,780
discover is that we can complete
the sort of strange

861
00:45:52,780 --> 00:45:56,400
business that we did at the
beginning of the course--

862
00:45:56,400 --> 00:45:57,650
or cosine x--

863
00:45:57,650 --> 00:46:01,010

864
00:46:01,010 --> 00:46:04,950
where we took the linear and
quadratic approximations.

865
00:46:04,950 --> 00:46:06,330
Now we're going to get complete

866
00:46:06,330 --> 00:46:10,060
formulas for these functions.

867
00:46:10,060 --> 00:46:15,180
Sine x turns out to be equal
to x minus x-cubed over 3

868
00:46:15,180 --> 00:46:20,510
factorial plus x to the 5th over
5 factorial minus x to

869
00:46:20,510 --> 00:46:24,860
the 7th over 7 factorial,
et cetera.

870
00:46:24,860 --> 00:46:29,510
And cosine x is equal to 1,
minus x-squared over 2

871
00:46:29,510 --> 00:46:30,680
factorial--

872
00:46:30,680 --> 00:46:33,400
that's the same as
this 2 here--

873
00:46:33,400 --> 00:46:39,030
plus x to the 4th over 4
factorial minus x to the 6th

874
00:46:39,030 --> 00:46:42,590
over 6 factorial,
plus et cetera.

875
00:46:42,590 --> 00:46:47,570
Now these may feel like they're
hard to memorize

876
00:46:47,570 --> 00:46:49,425
because I've just pulled
them out of a hat.

877
00:46:49,425 --> 00:46:51,990

878
00:46:51,990 --> 00:46:55,630
I do expect you to know them.

879
00:46:55,630 --> 00:46:58,990
They're actually extremely
similar formulas.

880
00:46:58,990 --> 00:47:02,230
The exponential here just has
this collection of factorials.

881
00:47:02,230 --> 00:47:06,860
The sine is all the odd powers
with alternating sines.

882
00:47:06,860 --> 00:47:10,260
And the cosine is all the even
powers with alternating sines.

883
00:47:10,260 --> 00:47:14,130
So all three of them form
part of the same family.

884
00:47:14,130 --> 00:47:16,900
So this will actually make it
easier for you to remember,

885
00:47:16,900 --> 00:47:18,150
rather than harder.

886
00:47:18,150 --> 00:47:21,490

887
00:47:21,490 --> 00:47:24,390
And so with that, I'll
leave the practice on

888
00:47:24,390 --> 00:47:26,210
differentiation for next time.

889
00:47:26,210 --> 00:47:27,570
And good luck, everybody.

890
00:47:27,570 --> 00:47:29,940
I'll talk to you individually.

891
00:47:29,940 --> 00:47:30,585